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Horizontal distance ( x metres) Height ( y metres) Time ( t seconds) Height ( y metres) Time ( t seconds) Horizontal distance ( x metres) Think about What assumptions are being made if the ball is modelled as a particle? Think about Which feature of a distance-time graph represents speed? Motion of a ball
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© Nuffield Foundation 2011
Nuffield Free-Standing Mathematics Activity
Galileo’s projectile
model
Galileo’s projectile model
How far will the ski jumper travel before he lands?
How can you model the motion of the ski jumper?
Horizontal distance(x metres)
Height(y metres)
Time(t seconds)
Height(y metres)
Time(t seconds)
Horizontal distance(x metres)
Think about What assumptions are being made if the ball is modelled as a particle?
Think about Which feature of a distance-time graph represents speed?
Motion of a ball
Galileo’s projectile model
x
y
Horizontal direction – the motion has constant speed.
Vertical direction – projectile accelerates at 9.8 ms–2.
Vertical distance fallen is proportional to t2.
aob c d e
f
h
i
g
l
n
Think about What can you say about bc, cd, and de?What does this tell you about the horizontal velocity of the ball and the horizontal distance covered by the ball?How could you check that the vertical distances are proportional to t2?
The modelling cycle
Defineproblem
Observe
Validate
Analyse
InterpretPredict
Real world MathematicsSet up a model
Experiment to validate Galileo’s model
Assumptions
• the ball is a particle
You need:
height h metres
range R metres
A B C
• air resistance is negligible
• the path of the projectile lies in a plane
Think about What modelling assumptions will be made?
Constants• the horizontal velocity of the
projectile after its launch from C
• the acceleration is g downwardsR
h
B C
Variables• the time, t seconds, measured from the instant of launch
• the height of the table h metres
• the distance, R metres, the ball lands from the foot of the table
Set up a model Think about What are the constants and variables?
Experiment to validate Galileo’s model
AnalyseUse the equations for motion in a straight line with constant acceleration to predict :
• how long it will take the ball to fall to the ground• the horizontal distance, R metres, it will have travelled
Practical advice
To estimate the velocity of the ball at launch:
• assume the ball has constant velocity along BC.
• time the ball travelling a measured distance along BC.
• calculate the average velocity from distance travelledtime taken
Vary the release point A to vary the launch velocity
Use talcum powder or salt on paper to find where the ball lands
Experiment to validate Galileo’s model
Investigate how theoretical predictions compare with experimental results
Why might there be discrepancies between the two graphs?
Interpret
RangeR metres
Velocity of projection u ms–1
Graph based on analysis Graph of experimental results
RangeR metres
Velocity of projection u ms–1
Experiment to validate Galileo’s model
Experiment to validate Galileo’s model
Graph based on analysis
RangeR metres
Velocity of projection u ms-1
221atuts
Vertical motion downwards
gives 24.9th
4.9ht
Horizontal motion
221atuts
R = utgives
4.9huR
4.9hgradient
Analyse
Extension: projection at an angle to the horizontal
x
y
O uhoriz
uvert vhoriz
vvert
(x, y)
at time t
atuv
221atuts
Find equations for vhoriz, x, vvert and y in terms of uhoriz, uvert, t
In the horizontal direction, a = 0
In the vertical direction, a = –9.8
Sketch graphs of vhoriz, x, vvert and y against t
x = uhoriz t
x
t0
vhoriz = uhoriz
uhoriz
vhoriz
t0
atuv
221atuts
In the horizontal direction, a = 0
Galileo’s projectile model
vvert = uvert – 9.8t
0t
vvert
uvert
y = uvert t – 4.9t2
t
y
0
atuv
221atuts
In the vertical direction, a = –9.8
Galileo’s projectile model
Reflect on your work
• What are the advantages of Galileo’s projectile model?
• Do your experimental results validate the model?
• Suggest some examples of motion which could not be modelled very well as projectiles.